Seminars and Colloquia by Series

TBA by Panos Stinis

Series
Applied and Computational Mathematics Seminar
Time
Monday, November 6, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005 and https://gatech.zoom.us/j/98355006347 (to be confirmed)
Speaker
Dr. Panos StinisPacific Northwest National Laboratory

Please Note: Speaker will present in person

TBA

Functional Ito Calculus for Lévy Processes (with a View Towards Mathematical Finance)

Series
Dissertation Defense
Time
Thursday, June 22, 2023 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006/Zoom
Speaker
Jorge Aurelio Víquez BolañosGeorgia Tech

Zoom link.  Meeting ID: 914 2801 6313, Passcode: 501018

We examine the relationship between Dupire's functional derivative and a variant of the functional derivative developed by Kim for analyzing functionals in systems with delay. Our findings demonstrate that if Dupire's space derivatives exist, differentiability in any continuous functional direction implies differentiability in any other direction, including the constant one. Additionally, we establish that co-invariant differentiable functionals can lead to a functional Ito formula in the Cont and Fournié path-wise setting under the right regularity conditions.

Next, our attention turns to functional extensions of the Meyer-Tanaka formula and the efforts made to characterize the zero-energy term for integral representations of functionals of semimartingales. Using Eisenbaum's idea for reversible semimartingales, we obtain an optimal integration formula for Lévy processes, which avoids imposing additional regularity requirements on the functional's space derivative, and extends other approaches using the stationary and martingale properties of Lévy processes.

Finally, we address the topic of integral representations for the delta of a path-dependent pay-off, which generalizes Benth, Di Nunno, and Khedher's framework for the approximation of functionals of jump-diffusions to cases where they may be driven by a process satisfying a path-dependent differential equation. Our results extend Jazaerli and Saporito's formula for the delta of functionals to the jump-diffusion case. We propose an adjoint formula for the horizontal derivative, hoping to obtain more tractable formulas for the Delta of strongly path-dependent functionals.

Committee 

  • Prof. Christian Houdré - School of Mathematics, Georgia Tech (advisor)
  • Prof. Michael Damron - School of Mathematics, Georgia Tech
  • Prof. Rachel Kuske - School of Mathematics, Georgia Tech
  • Prof. Andrzej Święch - School of Mathematics, Georgia Tech
  • Prof. José Figueroa-López - Department of Mathematics and Statistics, Washington University in St. Louis
  • Prof. Bruno Dupire - Department of Mathematics, New York University

Divisors and multiplicities under tropical and signed shadows

Series
Dissertation Defense
Time
Tuesday, June 20, 2023 - 09:30 for 1.5 hours (actually 80 minutes)
Location
Skiles 006 / Zoom
Speaker
Trevor GunnGeorgia Tech

Zoom link (Meeting ID: 941 5991 7033, Passcode: 328576)

I will present two projects related to tropical divisors and multiplicities. First, my work with Philipp Jell on fully-faithful tropicalizations in 3-dimensions. Second, my work with Andreas Gross on algebraic and combinatorial multiplicities for multivariate polynomials over the tropical and sign hyperfields.

The first part is about using piecewise linear functions to describe tropical curves in 3 dimensions and how the changes in those slopes (a divisor) lift to non-Archimedean curves. These divisors give an embedding of a curve in a 3-dimensional toric variety whose tropicalization is isometric to the so-called extended skeleton of the curve.

In part two, I describe how Baker and Lorscheid's theory of multiplicities over hyperfields can be extended to multivariate polynomials. One key result is a new proof/view of the work of Itenburg and Roy who used patchworking to construct some lower bounds on the number of positive roots of a system of polynomials given a particular sign arrangement. Another result is a collection of upper bounds for the same problem.

Committee:

  • Matt Baker (Advisor)
  • Josephine Yu
  • Oliver Lorscheid
  • Anton Leykin
  • Greg Blekherman

Improving and maximal inequalities in discrete harmonic analysis

Series
Dissertation Defense
Time
Wednesday, June 7, 2023 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 006 & online
Speaker
Christina GiannitsiGeorgia Tech

►Presentation will be in hybrid format. Zoom link: https://gatech.zoom.us/j/99128737217?pwd=dllnNE1kSW1DZURrY1UycGxrazJtQT09

►Abstract: We study various averaging operators of discrete functions, inspired by number theory, in order to show they satisfy  $\ell^p$ improving and maximal bounds. The maximal bounds are obtained via sparse domination results for $p \in (1,2)$, which imply boundedness on $\ell ^p (w)$ for $p \in (1, \infty )$, for all weights $w$ in the Muckenhoupt $A_p$ class. 

We start by looking at averages along the integers weighted by the divisor function $d(n)$, and obtain a uniform, scale free $\ell^p$-improving estimate for $p \in (1,2)$. We also show that the associated maximal function satisfies $(p,p)$ sparse bounds for $p \in (1,2)$. We move on to study averages along primes in arithmetic progressions, and establish improving and maximal inequalities for these averages, that are uniform in the choice of progression. The uniformity over progressions imposes several novel elements on our approach. Lastly, we generalize our setting in the context of number fields, by considering averages over the Gaussian primes.

Finally, we explore the connections of our work to number theory:   Fix an interval $\omega \subset \mathbb{T}$. There is an integer $N_\omega $, so that every odd integer $n$ with $N(n)>N_\omega $ is a sum of three Gaussian primes with arguments in $\omega $.  This is the weak Goldbach conjecture. A density version of the strong Goldbach conjecture is proved, as well.

                                                   

►Members of the committee:
· Michael Lacey (advisor)
· Chris Heil
· Ben Krause
· Doron Lubinsky
· Shahaf Nitzan

Two Phases of Scaling Laws for Nearest Neighbor Classifiers

Series
Applied and Computational Mathematics Seminar
Time
Thursday, May 25, 2023 - 10:30 for 1 hour (actually 50 minutes)
Location
https://gatech.zoom.us/j/98355006347
Speaker
Jingzhao ZhangTsinghua University

Please Note: Special time & day. Remote only.

A scaling law refers to the observation that the test performance of a model improves as the number of training data increases. A fast scaling law implies that one can solve machine learning problems by simply boosting the data and the model sizes. Yet, in many cases, the benefit of adding more data can be negligible. In this work, we study the rate of scaling laws of nearest neighbor classifiers. We show that a scaling law can have two phases: in the first phase, the generalization error depends polynomially on the data dimension and decreases fast; whereas in the second phase, the error depends exponentially on the data dimension and decreases slowly. Our analysis highlights the complexity of the data distribution in determining the generalization error. When the data distributes benignly, our result suggests that nearest neighbor classifier can achieve a generalization error that depends polynomially, instead of exponentially, on the data dimension.

Symmetric nonnegative polynomials and sums of squares: mean roads to infinity

Series
Dissertation Defense
Time
Wednesday, May 24, 2023 - 11:00 for 1.5 hours (actually 80 minutes)
Location
Skiles 006
Speaker
Jose AcevedoGeorgia Tech
We study the limits of cones of symmetric nonnegative polynomials and symmetric sums of squares of fixed degree, when expressed in power-mean or monomial-mean basis. These limits correspond to forms with stable expression in power-mean polynomials that are globally nonnegative (resp. sums of squares) regardless of the number of variables. Using some elements of the representation theory of the symmetric group we introduce partial symmetry reduction to describe the limit cone of symmetric sums of squares, which simultaneously allows us to tropicalize its dual cone. Using tropical convexity to describe the tropicalization of the dual cone to symmetric nonnegative forms we then compare both tropicalizations, which turn out to be convex polyhedral cones. We then show that the cones are different for all degrees larger than 4. For even symmetric forms we show that the cones agree up to degree $8$, and are different starting at degree 10. We also find, via tropicalization, explicit examples of symmetric forms that are nonnegative but not sums of squares at the limit.

Quantitative Generalized CLT with Self-Decomposable Limiting Laws by Spectral Methods

Series
Stochastics Seminar
Time
Thursday, May 18, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Benjamin ArrasUniversité de Lille

In this talk, I will present new stability results for non-degenerate centered self-decomposable laws with finite second moment and for non-degenerate symmetric alpha-stable laws with alpha in (1,2). These stability results are based on Stein's method and closed forms techniques. As an application, explicit rates of convergence are obtained for several instances of the generalized CLTs. Finally, I will discuss the standard Cauchy case.

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